Question: Solve for $x$ : $ 5|x - 5| + 8 = -2|x - 5| + 10 $
Explanation: Add $ {2|x - 5|} $ to both sides: $ \begin{eqnarray} 5|x - 5| + 8 &=& -2|x - 5| + 10 \\ \\ { + 2|x - 5|} && { + 2|x - 5|} \\ \\ 7|x - 5| + 8 &=& 10 \end{eqnarray} $ Subtract ${8}$ from both sides: $ \begin{eqnarray} 7|x - 5| + 8 &=& 10 \\ \\ { - 8} &=& { - 8} \\ \\ 7|x - 5| &=& 2 \end{eqnarray} $ Divide both sides by ${7}$ $ \dfrac{7|x - 5|} {{7}} = \dfrac{2} {{7}} $ Simplify: $ |x - 5| = \dfrac{2}{7}$ Because the absolute value of an expression is its distance from zero, it has two solutions, one negative and one positive: $ x - 5 = -\dfrac{2}{7} $ or $ x - 5 = \dfrac{2}{7} $ Solve for the solution where $x - 5$ is negative: $ x - 5 = -\dfrac{2}{7} $ Add ${5}$ to both sides: $ \begin{eqnarray} x - 5 &=& -\dfrac{2}{7} \\ \\ {+ 5} && {+ 5} \\ \\ x &=& -\dfrac{2}{7} + 5 \end{eqnarray} $ Change the ${ + 5}$ to an equivalent fraction with a denominator of $7$ $ x = - \dfrac{2}{7} {+ \dfrac{35}{7}} $ $ x = \dfrac{33}{7} $ Then calculate the solution where $x - 5$ is positive: $ x - 5 = \dfrac{2}{7} $ Add ${5}$ to both sides: $ \begin{eqnarray} x - 5 &=& \dfrac{2}{7} \\ \\ {+ 5} && {+ 5} \\ \\ x &=& \dfrac{2}{7} + 5 \end{eqnarray} $ Change the ${ + 5}$ to an equivalent fraction with a denominator of $7$ $ x = \dfrac{2}{7} {+ \dfrac{35}{7}} $ $ x = \dfrac{37}{7} $ Thus, the correct answer is $x = \dfrac{33}{7} $ or $x = \dfrac{37}{7} $.